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Notes on Contraction Theory a b Nicolas Tabareau Jean-Jacques Slotine aLaboratoire dePhysiologie delaPerceptionetdel’Action, CNRS-CollègedeFrance, 6 11placeMarcelinBerthelot, 75231ParisCedex05,France. 0 0 bNonlinear SystemLaboratory, Massachusetts InstituteofTechnology, 2 Cambridge,Massachusetts, 02139,USA n a J 6 Abstract ] O A Thesenotesderiveanumberoftechnicalresultsonnonlinear contraction theory,acom- . paratively recent tool for system stability analysis. In particular, they provide new results n on the preservation of contraction through system combinations, a property of interest in i l n modellingbiological systems. [ 2 Keywords: contraction theory,centralized feedback, hierarchies, attractors v 1 1 0 1 0 6 0 1 Introduction / n i l n Nonlinear contraction theory [LohmillerandSlotine,1998] is a comparatively re- : v cent tool for system stability analysis. These notes derive a number of technical i X resultsmotivatedbythetheory.Inparticular,theyprovidenewresultsonthepreser- r vation of contraction through system combinations, a property of interest in mod- a ellingbiologicalsystems. Section 2 analyzes the preservation of contraction through generalized negative feedbackbetweencontractingsystems.Section3describesanewsystemcombina- tion,centralizedcontraction,whichalsopreservescontractionbyaggregation.Sec- tion4usesstandardresultsfromcomputersciencetosimplifythegeneralstructure of arbitrary system combinations, and in particular to exploit intrinsic hierarchical properties. Section 5 discusses some applications to nonlinear attractors, while section 6 describes the estimation of the successive derivatives of a vector using compositevariables. correspondence to: N. Tabareau, CNRS, Laboratoire de Physiologie de la Perception ∗ et de l’Action, Collège de France, 11 place Marcellin Berthelot, 75231 Paris Cedex 05, France,,Tel.:+33-144271391, Fax:+33-144271382 E-mail:[email protected] PreprintsubmittedtoElsevierScience 2 Negativefeedback This section analyzes feedback connections which automatically give rise to con- tractionwithregardsto apredefined metric. Consider two contracting systems, of possibly different dimensions and metrics, andconnect theminfeedback, insuchaway thattheoverallvirtualdynamicsisof theform d δz1  F1 G(z,t)Bδz1 = − dt δz  G(z,t)T AT F δz   2  2  2 withA,Btwosquarematrices. Theoverallsystemiscontractingif (1) A andB aresymmetricpositivedefinite, and (2) thereexistsβ > 0 suchthat A˙ +A.F +F TA β A 1 1 B˙ +B.F +F TB ≤ −β B 2 2 ≤ − Indeed, wecan define themetric A 0   M =  0 B   Wehave d (δzTMδz) = MF+FTM+M˙ dt wherethematrixMF isoftheform,  AF1 AG(z,t)B MF = − BG(z,t)TA B.F   2  Thus d (δzTMδz) = δzT A˙ +AF1 +F1TA 0 δz βδzTMδz dt  0 B˙ +BF +F TB ≤ −  2 2  byhypothesis,whichimpliesthatδzTMδztendsexponentiallytozero.SinceMis positivedefinite, thisinturn impliesthatδzTδz tendsexponentiallytozero. 3 Centralized contraction We extend here the class of combinations of contracting systems described in [LohmillerandSlotine, 1998]. From a practical point of view, the condition given 2 for feedback combinations may be hard to deal with, whereas hierarchical combi- nationsaremuchsimplerbutnotgeneralenough.Itisthereforeofinteresttofinda combination having a strong expressiveness together with an automatic guarantee ofcontraction. An almost hierarchical feedback combination. The basic idea lies in the following remark. When looking at the combination depicted in figure 3, the loop between F and F seems to be illusory, as the domain and co-domain in F are 1 2 2 disjoint.Let’stry toformalizethisidea. We considertwo contracting systemconnecting in feedback in such a way that the second systemcan besplitintoz1 and z2 soto write 2 2 δz1  F1 G1 0 δz1 d δz1 =  0 F11 F12δz1 dt  2  2 2  2           δz2 G F21 F22δz2  2  2 2 2  2 Then, a following our idea that this almost represents a hierarchical combination, weapplythemetric I 0 0  Θ = 0 ǫ−1I 0        0 0 ǫI   Thisgivesrisetothegeneralized Jacobian Existing connection F 1 Forbidden connection 1 F 2 Co−Domain Domain F2 F2 2 Fig.1.Acombination thatseemstogiverisetoautomaticcontraction 3 δz1  F1 ǫG1 0 δz1 d δz1 =  0 F11 ǫ−2 F12δz1 dt  2  2 2  2           δz2 ǫ G ǫ2 F21 F22 δz2  2  2 2 2  2 This says that, as long as G and G are bounded, they are negligible. However, 1 2 wehavenomoreguaranteeonthecontractionofF asthematrixoffeedback F12 2 2 and F21 havebeen perturbed byǫ. 2 This tells us that we have to restrict this intuition to a particular kind of feedback withinF . 2 3.1 Orientablesystems The first step is to master the metric used in each local feedback. Indeed, as in the caseoffeedbackcombination,toapplythecombinationrecursively,weneedsome guarantees of non interference between the metric. Then idea is to require that the metric use for each combination only acts on the peripheral system and not on the centralizer. Thisleadsto thenotionoforientablesystem. Definition 1 Acombinationbetweentwosystemsissaidtobeorientableifametric thatmakes thegeneralizedJacobiannegativedefinitecan bewritten M′ 0   M =  0 I   Small gain. Consider two contracting systems, of possibly different dimensions and metrics, and connect them in feedback, in such a way that the overall virtual dynamicsisoftheform d δz1  F1 BG(z,t)δz1 = dt δz  G(z,t)T AT F δz   2  2  2 with A, B two square matrices. Note that in this form, A and B must have the samedimension. AssumenowthatA and Bsatisfy Bis invertible • AB−1 is constantandsymmetricpositivedefinite • 4 Wecan then definethemetric AB−1 0   M =  0 I   whichcan berewrittenM = ΘTΘwith √AB−1 0   Θ =  0 I   Usingthefact that √AB−1B = (√AB−1)−1A wehave √AB−1F1(√AB−1)−1 √AB−1BG(z,t)  G(z,t)T(√AB−1B)T F   2  Applying a standard result for small gain feedback (see [Slotine, 2003]), we can conclude on the contraction of the system if √AB−1F (√AB−1)−1 is negative 1 definiteand thefollowinginequalityholds σ2(√AB−1BG(z,t)) < λ((√AB−1F (√AB−1)−1) )λ((F ) ) (1) 1 s 2 s Wecan concludethatthissystemisan orientablescaling-robustsystem. Note that the aboveassumptionson A and B are verified in the common case that A isconstantand symmetricpositivedefiniteand B = λI withconstantλ > 0. Negativefeedback. In thesameway, forasystemoftheform d δz1  F1 BG(z,t)δz1 = − dt δz  G(z,t)T AT F δz   2  2  2 wecanconstructaorientablemetricsuchthatthesystemiscontractingif√AB−1F (√AB−1)−1 1 isnegativedefinite. 3.2 Orientablescaling-robustsystems Definition 2 A combination between two systems is said to be orientable scaling- I 0   robust if transformingthesystem by themetric D = (ǫ > 0) leads to an ǫ 0 ǫ I   5 M 0 orientablecombinationwithmetric ǫ . WefurtherrequirethatM ǫ→0 0. ǫ  0 I −−→   Remark 3 Thedefinitionabovesaysthatthedynamics d δx1 F11 ǫ−1 F12δx1 = dt δx2 ǫ F21 F22δx2      areorientableforallǫ. Smallgainandnegativefeedback. Letuscomebacktothetwopreviousexam- I 0   ples.ItisclearthatapplyingthemetricD = forǫ > 0leadstoanother ǫ 0 ǫI   contractingsystemwithmetric ǫAB−1 0   M = ǫ  0 I   So an orientable small gain (resp. negative feedback) is automatically scaling- robust. 3.3 Centralizedcontraction Assumethat wehave, as in figure 3.3, n systemsconnecting to a particularsystem calledthecenter insuchawaythateveryconnectiontothecenterisorientableand scaling-robust. Assume also that the connection between the different peripheral systemsis hierarchical. That kindofsystemcan berewritten δz F X X X X X F2 δz  6   6 6 6   δz   0 F X X X X F2 δz   5   5 5 5        δz4   0 0 F4 X X X F24 δz4  d       δz  =  0 0 0 F X X F2 δz  dt  3   3 3 3             δz   0 0 0 0 F X F2 δz   2   2 2 2        δz1   0 0 0 0 0 F1 F21 δz1       δz  F1 F1 F1 F1 F1 F1 C δz   C  6 5 4 3 2 1  C 6 1 6 ∼ ∼ 2 Center 5 ∼ ∼ ∼ ∼ 3 4 Fig.2.Acentralized combination ofcontracting systems Forthatparticularvirtualsystem,let ususethemetric I 0 0 0 0 0 0   0 ǫ−1I 0 0 0 0 0      0 0 ǫ−2I 0 0 0 0      0 0 0 ǫ−3I 0 0 0      0 0 0 0 ǫ−4I 0 0      0 0 0 0 0 ǫ−5I 0      0 0 0 0 0 0 ǫI   whichleads tothesystem δz F ǫX ǫ2X ǫ3X ǫ4X ǫ5X ǫ−1F2 δz  6   6 6 6   δz   0 F ǫX ǫ2X ǫ3X ǫ4X ǫ−2F2 δz   5   5 5 5        δz4   0 0 F4 ǫX ǫ2X ǫ3X ǫ−3F24 δz4  d       δz  =  0 0 0 F ǫX ǫ2X ǫ−4F2 δz  dt  3   3 3 3        δz   0 0 0 0 F ǫX ǫ−5F2 δz   2   2 2 2        δz1   0 0 0 0 0 F1 ǫ−6F21 δz1       δz  ǫF1 ǫ2F1 ǫ3F1 ǫ4F1 ǫ5F1 ǫ6F1 C δz   C  6 5 4 3 2 1  C 7 Since all the couple (F1,F2) are assumed to be orientable scaling-robust, for any i i smallǫ,thereisaconstantmetricMsuchthatthesymmetricpartofthegeneralized Jacobian can bewritten,whenǫ tendstozero, as H 0 0 0 0 0 K  6 6  0 H 0 0 0 0 K   5 5      0 0 H4 0 0 0 K4      0 0 0 H 0 0 K   3 3      0 0 0 0 H 0 K   2 2      0 0 0 0 0 H1 K1     KT KT KT KT KT KT C   6 5 4 3 2 1  wherethematrices H K  i i KT C   i  are allnegativedefinite. Applyingabasicresultofmatrixanalysisthusyieldsthecondition C < KTH−1K X i i i i whichis equivalentto λ(KTH−1K C−1) < 1 X i i i i A sufficientconditionis thus σ(K )2λ(H )−1 < λ(C) X i i i oreventhelessgeneral buteasier toverifyinequality σ(K )2 < λ(C)minλ(H ) X i i i i Wecall thiscasecentralizedcontraction. 3.4 Goingfurther Thestructureofcentralizedcontractionisageneralschemethatcanbeextendedto morecomplexstructures.We presenthere twobasicextensions. 8 Multiplelayers Wecanapplytheresultofcentralizedcontractionevenifthepe- ripheralsystemiscomposedofdifferentlayers.Forexample,thesystemdescribed in figure 3 is automatically contracting providing that the red connections are ori- entablescalingrobust. Fig.3.Centralized contraction formultiplelayers Multiple centers In biological systems, it is often of interest to consider a centralizer which is composed of multiple systems. In that case, the contraction can be guaranteed if all connections to the center considered as a whole are orientable scalingrobust. 4 Strongly connected components In computational neuroscience as in many biological fields, we have to deals with largesystems.Hereweexploitastandardalgorithmfromcomputerscience[Knuth,1997] todecomposealargesystemintosub-systems,inasuchwaythatthecontractionof theoverallsystemcanbededucedfromthecontractionofthesmallersub-systems. Definition 4 A strongly connected component of a directed graph G = (V,E) is a maximal set of vertices U V such that for all u,v U, u is reachable from v ⊂ ∈ andv isreachablefromu. Proposition5 Any directed graph is a union of strongly connected components plusedges tojointhecomponentstogether. Thus, we are able to distinguish between micro-systems which are connected in feedback combinationornot. Indeed,we can statetheproposition: 9 Proposition6 Two sub-systems of large systems are in feedback combination iff thosetwo systemsbelongsto thesamestronglyconnected component. Let usnowdescribethealgorithmtocomputesuch adecomposition. Algorithm. Strongly_connected_components(G) (1) Use the Depth-First-Search (DFS) algorithm to compute f[U] the finishing timeofu (2) ComputeGT = (V,E) where ET = (u,v) (v,u) E { | ∈ } (3) Execute DFS on GT by grabbing vertices in the order of decreasing f[u] as computedinstep 1. (4) Outputtheverticesofeachtreeinthedepth-firstforestofstep3.asaseparate stronglyconnected component Complexity. ThisalgorithmrunstwicethetimeofDFS(G)whichisΘ( V + E ) | | | | Fig.4.Thestrongly connected components ofalargesystem 10